In very rough terms, it's like being able to catch a ball - you watch it to see where it's going to go, and then put your hand there.

With a comet, measurements of its movement are taken, they plot out the path it takes, they determine how the gravitational effect of the Sun and planets will affect it, and then see if the estimated path intersects with where the Earth will be.

Yes, it's a bit complicated (computers help), and there are always things that can mess up the predictions, but that's the overview.

As a first approximation, it's just Newtonian physics: The force between two objects with mass M and m at distance r is F = GMm/r^(2) where G is the gravitational constant. If you calculate the acceleration of the object in the field of a different mass, then the mass of the object cancels: a = F/m = GM/r^(2). That means we don't need to know the comet mass (which can be difficult to estimate) to find its acceleration. Measure its position a few times to find its location and velocity and you can calculate its future trajectory step by step. If only the Sun's gravity is relevant then the comet will fly in an ellipse and you can use Kepler's laws. See if the trajectory intersects Earth's orbit, and see if both objects will be in the same place at the same time.

To refine the orbit estimates, you want to look at non-gravitational effects: Is the object ejecting gas because the Sun heats it up? Is there significant radiation pressure? Anything else? You can also use some corrections from relativistic effects compared to Newtonian physics. All that goes into better estimates how the object will accelerate, so it will improve future position estimates.

This is known as orbit determination.

https://en.wikipedia.org/wiki/Orbit_determination

A telescope can observe what direction something is in, but cannot directly measure the distance. But because an object's motion depends on its distance from other masses (mainly the sun), three observations at different times are sufficient to determine the orbit. Newton worked that out for the special case of a parabolic trajectory and later scientists refined the mathematics and extended it to all orbits.

The observations have error bars, which means the orbit determination also has error bars. Observations over a short period of time tend to result in less precise orbital information than observations spaced further apart, and extra observations are good to have. (And in some cases, we actually have radar distance measurements.) The error bars on the orbital information mean that a prediction of the future position of the comet is less precise the further into the future it is, and if the comet will pass a planet close that amplifies the errors. For small bodies like comets and asteroids non-gravitational effects, mainly relating to solar radiation, can perturb the orbit and they are hard to predict. All these factors mean don't usually know for certain that an asteroid or comet will hit or miss Earth, but instead you read things like a "1 in 100 chance of impact in 20xx".

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Advanced telescopes and radars detect and track celestial bodies and determine their trajectory. We pretty much perfectly mapped the gravitational effects of all the large bodies in our solar system so a computer calculates future trajectory with the data observatories gather. Calculations are relatively simple even a phone app can do it with right data inputs.

But problem is the ones we cant see. Radar is useful in close range. And optical telescopes require astroids to pass in front of a light source ie sun